TITLE: Hermite Differential Equation - Non-integer values of $\lambda$
QUESTION [6 upvotes]: The Hermite differential equation, given by :
$$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$
has solutions of the 
$$ y(x) = \mathcal{H_n(x)} $$ when $ \lambda \: \epsilon \:\mathcal{Z_+} $
Are there solutions to this equation for a more general case, where $ \lambda  $ is a real number ? More importantly are there convergent solutions for cases where  $ \lambda  $  is not integer.

REPLY [3 votes]: I suggest you have a look at
http://mathworld.wolfram.com/HermiteDifferentialEquation.html 
The general solution of this equation is
$$c_1 H_{\frac{\lambda }{2}}(x)+c_2 \, _1F_1\left(-\frac{\lambda
   }{4};\frac{1}{2};x^2\right)$$